I uninstalled Adium using "AppCleaner", which seems to have caused my "Internet Accounts" logins to malfunction (I'm guessing the uninstaller deleted necessary files that were modified by Adium). So, I don't want to use another uninstaller app to get rid of it.

I thought Adium was gone, but I now have an "Adium." Internet password in Keychain Access that reappears every time I delete it. Getting info on it tells me the location is "AdIM://Adium.".

Most of Adium is contained within a single application bundle—that's the green duck you dragged to your Applications folder in InstallingAdium.

Make sure Adium is not running.
Drag the Adium application (Usually located in /Applications) to the trash.

Removing settings, chat transcripts, and Xtras

Adium stores all of your important preferences (including accounts—specifically, the service each account is on and each account's username), all of your transcripts of previous chats (if you didn't turn logging off in Adium's Preferences), and all Xtras you've installed in this folder:

~/Library/Application Support/Adium 2.0/

Adium also places preferences and supporting files in the following locations (in your 'Home' folder):

Adium also installs Apple Address Book contextual menu actions via AppleScripts in the following location:

~/Library/Address Book Plug-Ins
Move all of these to the Trash (preferably with Address Book not running).

Removing saved passwords

Adium stores your passwords in your Keychain. To remove them from there open Keychain Access (in your Utilities folder)
search for “service.username” (for example, “AIM.jdoe”).
select it and press the Delete (⌫) key.
Keychain Access will require your keychain password to go through with this; usually, this is the same as your Mac OS X login password. (It is not the password you're deleting.)

That said, it is safer to leave them in the Keychain, so you don't delete a password saved by another application by mistake. There is no harm in having keychain items that you aren't using.